Heart disease is one of the major life-threatening diseases with high mortality and incidence worldwide. Several model systems, such as primary cells and animals, have been used to understand heart diseases and establish appropriate treatments. However, they have limitations in accuracy and reproducibility in recapitulating disease pathophysiology and evaluating drug responses. In recent years, three-dimensional (3D) cardiac tissue models produced using tissue engineering technology and human cells have outperformed conventional models. In particular, the integration of cell reprogramming techniques with bioengineering platforms (e.g., microfluidics, scaffolds, bioprinting, and biophysical stimuli) has facilitated the development of heart-on-a-chip, cardiac spheroid/organoid, and engineered heart tissue (EHT) to recapitulate the structural and functional features of the native human heart. These cardiac models have improved heart disease modeling and toxicological evaluation. In this review, we summarize the cell types for the fabrication of cardiac tissue models, introduce diverse 3D human cardiac tissue models, and discuss the strategies to enhance their complexity and maturity. Finally, recent studies in the modeling of various heart diseases are reviewed.